\label{sec:conclusion}

% implications of decisions, possibility of extension

% while not scalable for 100's of items, these represent
% the first general exact solution methods for capacitated
% multi-inventory control problems

% have to define exact solution and understand properties
% before proceeding to approximate as an extension of this
% work

% bilinear issues

% quadratic quite limited but useful for single continuous
% resource or continuous time

% symbolic constrained optimization -- other uses

% Future work:
%
% - Application to Inventory (makes clear that there are
%     simplifications possible in current solution), consider
%     submitting to CPAIOR, Informs?
% - Exponential Distribution for Stochasticity?
% - Parameterized linear programming, application to graphical
%     models or efficient LP solutions, MAP Bayesian approaches
%     for uniforms?
% - Affine XADD, useful for mixture of uniforms in Bayesian work.
% - Efficient linear programming -- exploiting factorized
%     structure in constraints using variable elimination
% - Symbolic Policy Iteration for CSA-MDPs (good for Inventory,
%     what other problems that have a simple policy?  Traffic?)
% - Theoretical guarantees on (linear) XADD (minimal function, 
%     canonical?)  Efficient LADD implementation.  Quadratic
%     linearization would need to come in max step.  Pruning
%     inside the LADD... implement feasibility checking.
% - Nonlinear solutions (can we solve any more expressive problems?)
% - Approximation
%   * adaptively linearize the problem (especially for nonlinear, 
%     bilinear)
%   * XADD approximation?  Othogonal polynomials?

We have presented an SDP solution for continuous state \emph{and}
action MDPs with the key contribution of \emph{symbolic constrained
optimization} to solve the continuous action maximization problem.  We
believe this is the first work to propose optimal closed-form
solutions to MDPs with \emph{multivariate} continuous state \emph{and}
actions, discrete noise, \emph{piecewise} linear dynamics, and
\emph{piecewise} linear (or restricted \emph{piecewise} quadratic)
reward; further, we believe our experimental results are the first
exact solutions to these problems to provide a closed-form optimal
policy for all (continuous) states.

